Contacts

Flow, vector field, and tensor field visualization

Introduction

Many phenomena in science cannot be represented adequately by a single value (or scalar). As soon as directional information is of importance, vectors or tensors are used. An example for this is the simulation of air in a closed room. While single values are enough to represent the temperature at every position in the room, vectors are needed to represent the flow of the air in the room. This example also shows the strong connection between vector field and flow visualization. Vector fields are usually the basis for flow visualization applications.

In more complex cases, where not only a single direction but also a directional distribution is given, even a vector is not sufficient. In this case, tensors are used. Tensors are the generalization of scalars and vectors. A scalar is a tensor of zeroth order and a vector is a tensor of first order.

Arrow glyphs visualizing the local flow direction of simulated air in a closed room.

Pathlines visualizing the path of massless particles in the air flow.

Vector field and flow visualization

A typical source for vector fields are computational fluid dynamics applications. In this case, the vector field describes the flow direction of the fluid. A standard technique for visualizing them is the usage of arrow glyphs showing the local flow direction. Another common method is the visualization of tangential curves, .e.g, path lines for time-dependent vector fields.

Both approaches have drawbacks in that they visualize only instantaneous local properties (glyphs) or are prone to visual clutter (path lines). For time-dependent vector fields, a new approach became popular, the visualization of Lagrangian coherent structures by using the finite-time Lyapunov (FTLE) exponent. The FTLE is a measure for the separation of particles after a given time. Low FTLE values indicate low separation and therefore coherent regions in the flow. High FTLE values indicate separation and occur between different coherent regions in the flow. Hence, the FTLE field is separating the flow field in regions of coherent behavior.

Computing the FTLE field is computationally expensive, because an integral curve has to be computed for every element of resulting field. We developed a hierarchical scheme to accelerate this computation. With our method, the computational complexity is reduced from linear to logarithmic. As a result, the computation of a time series of 3D FTLE fields can be done in a few minutes instead of several hours.

2D forward FTLE field of simulated air in a closed room.

3D forward FTLE field of the Arnold-Beltrami-Childress flow.

Tensor field visualization

Second order tensors are used to describe the directional distribution of properties. They are used, e.g., to describe diffusion or mechanical stress. Such data are generated e.g., with diffusion tensor magnetic resonance tomographie.

One approach to visualize these tensors is the usage of special glyphs, similar to arrow glyphs for vectors. However, only local information is directly visible in this case, relations between neighboring positions are only indirectly recognizable. Another common approach is the display of lines following the major eigenvector of the tensor. In this case, not only local properties are visualized. However, this approach is only feasible for low number of lines. Visualizing the complete dataset with this approach results in massive visual clutter.

A possible solution for both problems, visualizing not only local properties and avoiding visual clutter, is the extraction and visualization of structural information from the data. We therefore developed a method to visualize coherent structures in second-order tensor fields.

Visualization of coherent structures in a human brain.

Visualization of nerve tracts in the brain. Coherent regions are consistent with bundles of nerve fibers.